Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3

Percentage Accurate: 67.1% → 93.0%

Time: 12.2s

Alternatives: 18

Speedup: 0.9×

Specification

Math

\[\begin{array}{l} \\ x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y x) (- z t)) (- a t))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (((y - x) * (z - t)) / (a - t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

Python

def code(x, y, z, t, a):
	return x + (((y - x) * (z - t)) / (a - t))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (((y - x) * (z - t)) / (a - t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 67.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (/ (* (- y x) (- z t)) (- a t))))

C

double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (((y - x) * (z - t)) / (a - t))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (((y - x) * (z - t)) / (a - t));
}

Python

def code(x, y, z, t, a):
	return x + (((y - x) * (z - t)) / (a - t))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (((y - x) * (z - t)) / (a - t));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 93.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - x, \frac{z}{a - t}, \mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\right)\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-298}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;y + \frac{\left(y - x\right) \cdot \left(a - z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y x) (/ z (- a t)) (fma (- x y) (/ t (- a t)) x)))
        (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 -1e-298)
     t_1
     (if (<= t_2 0.0) (+ y (/ (* (- y x) (- a z)) t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - x), (z / (a - t)), fma((x - y), (t / (a - t)), x));
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -1e-298) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = y + (((y - x) * (a - z)) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - x), Float64(z / Float64(a - t)), fma(Float64(x - y), Float64(t / Float64(a - t)), x))
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -1e-298)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(y + Float64(Float64(Float64(y - x) * Float64(a - z)) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] + N[(N[(x - y), $MachinePrecision] * N[(t / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-298], t$95$1, If[LessEqual[t$95$2, 0.0], N[(y + N[(N[(N[(y - x), $MachinePrecision] * N[(a - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -9.99999999999999912e-299 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 72.3%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right) + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      3. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot z}}{a - t} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{\color{blue}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{a - t}, \color{blue}{-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + x}\right) \]

   5. Applied rewrites 90.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, \mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\right)} \]

   if -9.99999999999999912e-299 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right) + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      3. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot z}}{a - t} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{\color{blue}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{a - t}, \color{blue}{-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + x}\right) \]

   5. Applied rewrites 46.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, \mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. associate-*r/ N/A

         \[\leadsto y + \left(\color{blue}{\frac{-1 \cdot \left(z \cdot \left(y - x\right)\right)}{t}} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right) \]

      3. associate-*r/ N/A

         \[\leadsto y + \left(\frac{-1 \cdot \left(z \cdot \left(y - x\right)\right)}{t} - \color{blue}{\frac{-1 \cdot \left(a \cdot \left(y - x\right)\right)}{t}}\right) \]

      4. mul-1-neg N/A

         \[\leadsto y + \left(\frac{-1 \cdot \left(z \cdot \left(y - x\right)\right)}{t} - \frac{\color{blue}{\mathsf{neg}\left(a \cdot \left(y - x\right)\right)}}{t}\right) \]

      5. div-sub N/A

         \[\leadsto y + \color{blue}{\frac{-1 \cdot \left(z \cdot \left(y - x\right)\right) - \left(\mathsf{neg}\left(a \cdot \left(y - x\right)\right)\right)}{t}} \]

      6. mul-1-neg N/A

         \[\leadsto y + \frac{-1 \cdot \left(z \cdot \left(y - x\right)\right) - \color{blue}{-1 \cdot \left(a \cdot \left(y - x\right)\right)}}{t} \]

      7. distribute-lft-out-- N/A

         \[\leadsto y + \frac{\color{blue}{-1 \cdot \left(z \cdot \left(y - x\right) - a \cdot \left(y - x\right)\right)}}{t} \]

      8. associate-*r/ N/A

         \[\leadsto y + \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      9. mul-1-neg N/A

         \[\leadsto y + \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} \]

      10. unsub-neg N/A

          \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      11. lower--.f64 N/A

          \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      12. lower-/.f64 N/A

          \[\leadsto y - \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{y - \frac{\left(y - x\right) \cdot \left(z - a\right)}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -1 \cdot 10^{-298}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a - t}, \mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0:\\ \;\;\;\;y + \frac{\left(y - x\right) \cdot \left(a - z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{a - t}, \mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - x}{\frac{a - t}{z - t}}\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-298}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;y + \frac{\left(y - x\right) \cdot \left(a - z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (- y x) (/ (- a t) (- z t)))))
        (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 -1e-298)
     t_1
     (if (<= t_2 0.0) (+ y (/ (* (- y x) (- a z)) t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - x) / ((a - t) / (z - t)));
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -1e-298) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = y + (((y - x) * (a - z)) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x + ((y - x) / ((a - t) / (z - t)))
    t_2 = x + (((y - x) * (z - t)) / (a - t))
    if (t_2 <= (-1d-298)) then
        tmp = t_1
    else if (t_2 <= 0.0d0) then
        tmp = y + (((y - x) * (a - z)) / t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - x) / ((a - t) / (z - t)));
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -1e-298) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = y + (((y - x) * (a - z)) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + ((y - x) / ((a - t) / (z - t)))
	t_2 = x + (((y - x) * (z - t)) / (a - t))
	tmp = 0
	if t_2 <= -1e-298:
		tmp = t_1
	elif t_2 <= 0.0:
		tmp = y + (((y - x) * (a - z)) / t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y - x) / Float64(Float64(a - t) / Float64(z - t))))
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -1e-298)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(y + Float64(Float64(Float64(y - x) * Float64(a - z)) / t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + ((y - x) / ((a - t) / (z - t)));
	t_2 = x + (((y - x) * (z - t)) / (a - t));
	tmp = 0.0;
	if (t_2 <= -1e-298)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = y + (((y - x) * (a - z)) / t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - x), $MachinePrecision] / N[(N[(a - t), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-298], t$95$1, If[LessEqual[t$95$2, 0.0], N[(y + N[(N[(N[(y - x), $MachinePrecision] * N[(a - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -9.99999999999999912e-299 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 72.3%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. lift-*.f64 N/A

         \[\leadsto x + \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a - t} \]

      3. associate-/l* N/A

         \[\leadsto x + \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} \]

      4. clear-num N/A

         \[\leadsto x + \left(y - x\right) \cdot \color{blue}{\frac{1}{\frac{a - t}{z - t}}} \]

      5. un-div-inv N/A

         \[\leadsto x + \color{blue}{\frac{y - x}{\frac{a - t}{z - t}}} \]

      6. lower-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{y - x}{\frac{a - t}{z - t}}} \]

      7. lower-/.f64 88.4

         \[\leadsto x + \frac{y - x}{\color{blue}{\frac{a - t}{z - t}}} \]

   4. Applied rewrites 88.4%

      \[\leadsto x + \color{blue}{\frac{y - x}{\frac{a - t}{z - t}}} \]

   if -9.99999999999999912e-299 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right) + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      3. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot z}}{a - t} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{\color{blue}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{a - t}, \color{blue}{-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + x}\right) \]

   5. Applied rewrites 46.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, \mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. associate-*r/ N/A

         \[\leadsto y + \left(\color{blue}{\frac{-1 \cdot \left(z \cdot \left(y - x\right)\right)}{t}} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right) \]

      3. associate-*r/ N/A

         \[\leadsto y + \left(\frac{-1 \cdot \left(z \cdot \left(y - x\right)\right)}{t} - \color{blue}{\frac{-1 \cdot \left(a \cdot \left(y - x\right)\right)}{t}}\right) \]

      4. mul-1-neg N/A

         \[\leadsto y + \left(\frac{-1 \cdot \left(z \cdot \left(y - x\right)\right)}{t} - \frac{\color{blue}{\mathsf{neg}\left(a \cdot \left(y - x\right)\right)}}{t}\right) \]

      5. div-sub N/A

         \[\leadsto y + \color{blue}{\frac{-1 \cdot \left(z \cdot \left(y - x\right)\right) - \left(\mathsf{neg}\left(a \cdot \left(y - x\right)\right)\right)}{t}} \]

      6. mul-1-neg N/A

         \[\leadsto y + \frac{-1 \cdot \left(z \cdot \left(y - x\right)\right) - \color{blue}{-1 \cdot \left(a \cdot \left(y - x\right)\right)}}{t} \]

      7. distribute-lft-out-- N/A

         \[\leadsto y + \frac{\color{blue}{-1 \cdot \left(z \cdot \left(y - x\right) - a \cdot \left(y - x\right)\right)}}{t} \]

      8. associate-*r/ N/A

         \[\leadsto y + \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      9. mul-1-neg N/A

         \[\leadsto y + \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} \]

      10. unsub-neg N/A

          \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      11. lower--.f64 N/A

          \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      12. lower-/.f64 N/A

          \[\leadsto y - \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{y - \frac{\left(y - x\right) \cdot \left(z - a\right)}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -1 \cdot 10^{-298}:\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z - t}}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0:\\ \;\;\;\;y + \frac{\left(y - x\right) \cdot \left(a - z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z - t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-298}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;y + \frac{\left(y - x\right) \cdot \left(a - z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z t) (- a t)) (- y x) x))
        (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 -1e-298)
     t_1
     (if (<= t_2 0.0) (+ y (/ (* (- y x) (- a z)) t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - t) / (a - t)), (y - x), x);
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -1e-298) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = y + (((y - x) * (a - z)) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - t) / Float64(a - t)), Float64(y - x), x)
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -1e-298)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = Float64(y + Float64(Float64(Float64(y - x) * Float64(a - z)) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-298], t$95$1, If[LessEqual[t$95$2, 0.0], N[(y + N[(N[(N[(y - x), $MachinePrecision] * N[(a - z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -9.99999999999999912e-299 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 72.3%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} + x} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}} + x \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot \left(z - t\right)}}{a - t} + x \]

      5. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z - t}{a - t}} + x \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - t}{a - t} \cdot \left(y - x\right)} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)} \]

      8. lower-/.f64 88.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - t}{a - t}}, y - x, x\right) \]

   4. Applied rewrites 88.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)} \]

   if -9.99999999999999912e-299 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right) + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      3. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot z}}{a - t} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{\color{blue}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{a - t}, \color{blue}{-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + x}\right) \]

   5. Applied rewrites 46.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, \mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   7. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. associate-*r/ N/A

         \[\leadsto y + \left(\color{blue}{\frac{-1 \cdot \left(z \cdot \left(y - x\right)\right)}{t}} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right) \]

      3. associate-*r/ N/A

         \[\leadsto y + \left(\frac{-1 \cdot \left(z \cdot \left(y - x\right)\right)}{t} - \color{blue}{\frac{-1 \cdot \left(a \cdot \left(y - x\right)\right)}{t}}\right) \]

      4. mul-1-neg N/A

         \[\leadsto y + \left(\frac{-1 \cdot \left(z \cdot \left(y - x\right)\right)}{t} - \frac{\color{blue}{\mathsf{neg}\left(a \cdot \left(y - x\right)\right)}}{t}\right) \]

      5. div-sub N/A

         \[\leadsto y + \color{blue}{\frac{-1 \cdot \left(z \cdot \left(y - x\right)\right) - \left(\mathsf{neg}\left(a \cdot \left(y - x\right)\right)\right)}{t}} \]

      6. mul-1-neg N/A

         \[\leadsto y + \frac{-1 \cdot \left(z \cdot \left(y - x\right)\right) - \color{blue}{-1 \cdot \left(a \cdot \left(y - x\right)\right)}}{t} \]

      7. distribute-lft-out-- N/A

         \[\leadsto y + \frac{\color{blue}{-1 \cdot \left(z \cdot \left(y - x\right) - a \cdot \left(y - x\right)\right)}}{t} \]

      8. associate-*r/ N/A

         \[\leadsto y + \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      9. mul-1-neg N/A

         \[\leadsto y + \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} \]

      10. unsub-neg N/A

          \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      11. lower--.f64 N/A

          \[\leadsto \color{blue}{y - \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      12. lower-/.f64 N/A

          \[\leadsto y - \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

   8. Applied rewrites 99.8%

      \[\leadsto \color{blue}{y - \frac{\left(y - x\right) \cdot \left(z - a\right)}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq -1 \cdot 10^{-298}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \leq 0:\\ \;\;\;\;y + \frac{\left(y - x\right) \cdot \left(a - z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 37.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+115}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq -9.5 \cdot 10^{-124}:\\ \;\;\;\;z \cdot \frac{x - y}{t}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-70}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+109}:\\ \;\;\;\;x \cdot \frac{z - a}{t}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7.5e+115)
   y
   (if (<= t -9.5e-124)
     (* z (/ (- x y) t))
     (if (<= t 1.3e-70)
       (* y (/ z a))
       (if (<= t 2.2e+109) (* x (/ (- z a) t)) y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7.5e+115) {
		tmp = y;
	} else if (t <= -9.5e-124) {
		tmp = z * ((x - y) / t);
	} else if (t <= 1.3e-70) {
		tmp = y * (z / a);
	} else if (t <= 2.2e+109) {
		tmp = x * ((z - a) / t);
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-7.5d+115)) then
        tmp = y
    else if (t <= (-9.5d-124)) then
        tmp = z * ((x - y) / t)
    else if (t <= 1.3d-70) then
        tmp = y * (z / a)
    else if (t <= 2.2d+109) then
        tmp = x * ((z - a) / t)
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7.5e+115) {
		tmp = y;
	} else if (t <= -9.5e-124) {
		tmp = z * ((x - y) / t);
	} else if (t <= 1.3e-70) {
		tmp = y * (z / a);
	} else if (t <= 2.2e+109) {
		tmp = x * ((z - a) / t);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -7.5e+115:
		tmp = y
	elif t <= -9.5e-124:
		tmp = z * ((x - y) / t)
	elif t <= 1.3e-70:
		tmp = y * (z / a)
	elif t <= 2.2e+109:
		tmp = x * ((z - a) / t)
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -7.5e+115)
		tmp = y;
	elseif (t <= -9.5e-124)
		tmp = Float64(z * Float64(Float64(x - y) / t));
	elseif (t <= 1.3e-70)
		tmp = Float64(y * Float64(z / a));
	elseif (t <= 2.2e+109)
		tmp = Float64(x * Float64(Float64(z - a) / t));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -7.5e+115)
		tmp = y;
	elseif (t <= -9.5e-124)
		tmp = z * ((x - y) / t);
	elseif (t <= 1.3e-70)
		tmp = y * (z / a);
	elseif (t <= 2.2e+109)
		tmp = x * ((z - a) / t);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -7.5e+115], y, If[LessEqual[t, -9.5e-124], N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.3e-70], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.2e+109], N[(x * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], y]]]]

Derivation

1. Split input into 4 regimes

2. if t < -7.4999999999999997e115 or 2.1999999999999999e109 < t

   1. Initial program 39.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right) + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      3. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot z}}{a - t} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{\color{blue}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{a - t}, \color{blue}{-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + x}\right) \]

   5. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, \mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \left(x + -1 \cdot x\right) - \color{blue}{-1 \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.7%

      \[\leadsto y \]

   if -7.4999999999999997e115 < t < -9.49999999999999989e-124

   1. Initial program 78.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto z \cdot \color{blue}{\left(\frac{x}{t} - \frac{y}{t}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.3%

      \[\leadsto z \cdot \color{blue}{\frac{x - y}{t}} \]

   if -9.49999999999999989e-124 < t < 1.30000000000000001e-70

   1. Initial program 88.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{a - t} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{z \cdot \color{blue}{\left(y - x\right)}}{a - t} \]

      6. lower--.f64 48.1

         \[\leadsto \frac{z \cdot \left(y - x\right)}{\color{blue}{a - t}} \]

   5. Applied rewrites 48.1%

      \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{z \cdot \left(y - x\right)}{\color{blue}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.7%

      \[\leadsto z \cdot \color{blue}{\frac{y - x}{a}} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{y \cdot z}{a} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.8%

       \[\leadsto y \cdot \frac{z}{\color{blue}{a}} \]

   if 1.30000000000000001e-70 < t < 2.1999999999999999e109

   1. Initial program 62.6%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{z}{t} - \frac{a}{t}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 35.5%

      \[\leadsto \frac{x \cdot \left(z - a\right)}{\color{blue}{t}} \]
   9. Step-by-step derivation

   10. Applied rewrites 39.5%

       \[\leadsto \frac{z - a}{t} \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+115}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq -9.5 \cdot 10^{-124}:\\ \;\;\;\;z \cdot \frac{x - y}{t}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-70}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+109}:\\ \;\;\;\;x \cdot \frac{z - a}{t}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 37.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+115}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq -9.5 \cdot 10^{-124}:\\ \;\;\;\;z \cdot \frac{x - y}{t}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-70}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+109}:\\ \;\;\;\;\left(z - a\right) \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7.5e+115)
   y
   (if (<= t -9.5e-124)
     (* z (/ (- x y) t))
     (if (<= t 1.3e-70)
       (* y (/ z a))
       (if (<= t 2.2e+109) (* (- z a) (/ x t)) y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7.5e+115) {
		tmp = y;
	} else if (t <= -9.5e-124) {
		tmp = z * ((x - y) / t);
	} else if (t <= 1.3e-70) {
		tmp = y * (z / a);
	} else if (t <= 2.2e+109) {
		tmp = (z - a) * (x / t);
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-7.5d+115)) then
        tmp = y
    else if (t <= (-9.5d-124)) then
        tmp = z * ((x - y) / t)
    else if (t <= 1.3d-70) then
        tmp = y * (z / a)
    else if (t <= 2.2d+109) then
        tmp = (z - a) * (x / t)
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7.5e+115) {
		tmp = y;
	} else if (t <= -9.5e-124) {
		tmp = z * ((x - y) / t);
	} else if (t <= 1.3e-70) {
		tmp = y * (z / a);
	} else if (t <= 2.2e+109) {
		tmp = (z - a) * (x / t);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -7.5e+115:
		tmp = y
	elif t <= -9.5e-124:
		tmp = z * ((x - y) / t)
	elif t <= 1.3e-70:
		tmp = y * (z / a)
	elif t <= 2.2e+109:
		tmp = (z - a) * (x / t)
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -7.5e+115)
		tmp = y;
	elseif (t <= -9.5e-124)
		tmp = Float64(z * Float64(Float64(x - y) / t));
	elseif (t <= 1.3e-70)
		tmp = Float64(y * Float64(z / a));
	elseif (t <= 2.2e+109)
		tmp = Float64(Float64(z - a) * Float64(x / t));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -7.5e+115)
		tmp = y;
	elseif (t <= -9.5e-124)
		tmp = z * ((x - y) / t);
	elseif (t <= 1.3e-70)
		tmp = y * (z / a);
	elseif (t <= 2.2e+109)
		tmp = (z - a) * (x / t);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -7.5e+115], y, If[LessEqual[t, -9.5e-124], N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.3e-70], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.2e+109], N[(N[(z - a), $MachinePrecision] * N[(x / t), $MachinePrecision]), $MachinePrecision], y]]]]

Derivation

1. Split input into 4 regimes

2. if t < -7.4999999999999997e115 or 2.1999999999999999e109 < t

   1. Initial program 39.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right) + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      3. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot z}}{a - t} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{\color{blue}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{a - t}, \color{blue}{-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + x}\right) \]

   5. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, \mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \left(x + -1 \cdot x\right) - \color{blue}{-1 \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.7%

      \[\leadsto y \]

   if -7.4999999999999997e115 < t < -9.49999999999999989e-124

   1. Initial program 78.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Applied rewrites 59.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto z \cdot \color{blue}{\left(\frac{x}{t} - \frac{y}{t}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.3%

      \[\leadsto z \cdot \color{blue}{\frac{x - y}{t}} \]

   if -9.49999999999999989e-124 < t < 1.30000000000000001e-70

   1. Initial program 88.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{a - t} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{z \cdot \color{blue}{\left(y - x\right)}}{a - t} \]

      6. lower--.f64 48.1

         \[\leadsto \frac{z \cdot \left(y - x\right)}{\color{blue}{a - t}} \]

   5. Applied rewrites 48.1%

      \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{z \cdot \left(y - x\right)}{\color{blue}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.7%

      \[\leadsto z \cdot \color{blue}{\frac{y - x}{a}} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{y \cdot z}{a} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.8%

       \[\leadsto y \cdot \frac{z}{\color{blue}{a}} \]

   if 1.30000000000000001e-70 < t < 2.1999999999999999e109

   1. Initial program 62.6%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{z}{t} - \frac{a}{t}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 35.5%

      \[\leadsto \frac{x \cdot \left(z - a\right)}{\color{blue}{t}} \]
   9. Step-by-step derivation

   10. Applied rewrites 39.4%

       \[\leadsto \left(z - a\right) \cdot \frac{x}{\color{blue}{t}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 39.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{x - y}{t}\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{+115}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq -9.5 \cdot 10^{-124}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-82}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* z (/ (- x y) t))))
   (if (<= t -7.5e+115)
     y
     (if (<= t -9.5e-124)
       t_1
       (if (<= t 2e-82) (* y (/ z a)) (if (<= t 2.2e+109) t_1 y))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = z * ((x - y) / t);
	double tmp;
	if (t <= -7.5e+115) {
		tmp = y;
	} else if (t <= -9.5e-124) {
		tmp = t_1;
	} else if (t <= 2e-82) {
		tmp = y * (z / a);
	} else if (t <= 2.2e+109) {
		tmp = t_1;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * ((x - y) / t)
    if (t <= (-7.5d+115)) then
        tmp = y
    else if (t <= (-9.5d-124)) then
        tmp = t_1
    else if (t <= 2d-82) then
        tmp = y * (z / a)
    else if (t <= 2.2d+109) then
        tmp = t_1
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = z * ((x - y) / t);
	double tmp;
	if (t <= -7.5e+115) {
		tmp = y;
	} else if (t <= -9.5e-124) {
		tmp = t_1;
	} else if (t <= 2e-82) {
		tmp = y * (z / a);
	} else if (t <= 2.2e+109) {
		tmp = t_1;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = z * ((x - y) / t)
	tmp = 0
	if t <= -7.5e+115:
		tmp = y
	elif t <= -9.5e-124:
		tmp = t_1
	elif t <= 2e-82:
		tmp = y * (z / a)
	elif t <= 2.2e+109:
		tmp = t_1
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(z * Float64(Float64(x - y) / t))
	tmp = 0.0
	if (t <= -7.5e+115)
		tmp = y;
	elseif (t <= -9.5e-124)
		tmp = t_1;
	elseif (t <= 2e-82)
		tmp = Float64(y * Float64(z / a));
	elseif (t <= 2.2e+109)
		tmp = t_1;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = z * ((x - y) / t);
	tmp = 0.0;
	if (t <= -7.5e+115)
		tmp = y;
	elseif (t <= -9.5e-124)
		tmp = t_1;
	elseif (t <= 2e-82)
		tmp = y * (z / a);
	elseif (t <= 2.2e+109)
		tmp = t_1;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.5e+115], y, If[LessEqual[t, -9.5e-124], t$95$1, If[LessEqual[t, 2e-82], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.2e+109], t$95$1, y]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.4999999999999997e115 or 2.1999999999999999e109 < t

   1. Initial program 39.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right) + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      3. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot z}}{a - t} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{\color{blue}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{a - t}, \color{blue}{-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + x}\right) \]

   5. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, \mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \left(x + -1 \cdot x\right) - \color{blue}{-1 \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.7%

      \[\leadsto y \]

   if -7.4999999999999997e115 < t < -9.49999999999999989e-124 or 2e-82 < t < 2.1999999999999999e109

   1. Initial program 71.6%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Applied rewrites 61.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto z \cdot \color{blue}{\left(\frac{x}{t} - \frac{y}{t}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.1%

      \[\leadsto z \cdot \color{blue}{\frac{x - y}{t}} \]

   if -9.49999999999999989e-124 < t < 2e-82

   1. Initial program 88.1%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{a - t} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{z \cdot \color{blue}{\left(y - x\right)}}{a - t} \]

      6. lower--.f64 48.6

         \[\leadsto \frac{z \cdot \left(y - x\right)}{\color{blue}{a - t}} \]

   5. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{z \cdot \left(y - x\right)}{\color{blue}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.2%

      \[\leadsto z \cdot \color{blue}{\frac{y - x}{a}} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{y \cdot z}{a} \]
   10. Step-by-step derivation

   11. Applied rewrites 36.2%

       \[\leadsto y \cdot \frac{z}{\color{blue}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 46.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+115}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq -1.18 \cdot 10^{-40}:\\ \;\;\;\;z \cdot \frac{x - y}{t}\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{+82}:\\ \;\;\;\;x - \frac{x \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7.5e+115)
   y
   (if (<= t -1.18e-40)
     (* z (/ (- x y) t))
     (if (<= t 1.42e+82) (- x (/ (* x z) a)) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7.5e+115) {
		tmp = y;
	} else if (t <= -1.18e-40) {
		tmp = z * ((x - y) / t);
	} else if (t <= 1.42e+82) {
		tmp = x - ((x * z) / a);
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-7.5d+115)) then
        tmp = y
    else if (t <= (-1.18d-40)) then
        tmp = z * ((x - y) / t)
    else if (t <= 1.42d+82) then
        tmp = x - ((x * z) / a)
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7.5e+115) {
		tmp = y;
	} else if (t <= -1.18e-40) {
		tmp = z * ((x - y) / t);
	} else if (t <= 1.42e+82) {
		tmp = x - ((x * z) / a);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -7.5e+115:
		tmp = y
	elif t <= -1.18e-40:
		tmp = z * ((x - y) / t)
	elif t <= 1.42e+82:
		tmp = x - ((x * z) / a)
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -7.5e+115)
		tmp = y;
	elseif (t <= -1.18e-40)
		tmp = Float64(z * Float64(Float64(x - y) / t));
	elseif (t <= 1.42e+82)
		tmp = Float64(x - Float64(Float64(x * z) / a));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -7.5e+115)
		tmp = y;
	elseif (t <= -1.18e-40)
		tmp = z * ((x - y) / t);
	elseif (t <= 1.42e+82)
		tmp = x - ((x * z) / a);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -7.5e+115], y, If[LessEqual[t, -1.18e-40], N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.42e+82], N[(x - N[(N[(x * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], y]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.4999999999999997e115 or 1.41999999999999993e82 < t

   1. Initial program 40.3%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right) + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      3. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot z}}{a - t} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{\color{blue}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{a - t}, \color{blue}{-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + x}\right) \]

   5. Applied rewrites 77.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, \mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \left(x + -1 \cdot x\right) - \color{blue}{-1 \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.7%

      \[\leadsto y \]

   if -7.4999999999999997e115 < t < -1.1799999999999999e-40

   1. Initial program 70.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Applied rewrites 64.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto z \cdot \color{blue}{\left(\frac{x}{t} - \frac{y}{t}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.0%

      \[\leadsto z \cdot \color{blue}{\frac{x - y}{t}} \]

   if -1.1799999999999999e-40 < t < 1.41999999999999993e82

   1. Initial program 85.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{z - t}{a - t}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{z - t}{a - t} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{z - t}{a - t}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{z - t}{a - t}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(z - t\right)}{a - t}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(z - t\right) \cdot x}}{a - t}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z - t\right) \cdot \frac{x}{a - t}}\right)\right) + 1 \cdot x \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right)} + 1 \cdot x \]

      10. *-lft-identity N/A

          \[\leadsto \left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right) + \color{blue}{x} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z - t}, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right) \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}\right)}, x\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}\right)}, x\right) \]

      17. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t} + \left(\mathsf{neg}\left(a\right)\right)}, x\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

      20. lower-neg.f64 58.9

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{t + \color{blue}{\left(-a\right)}}, x\right) \]

   5. Applied rewrites 58.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{x}{t + \left(-a\right)}, x\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto x + \color{blue}{-1 \cdot \frac{x \cdot z}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 53.7%

      \[\leadsto x - \color{blue}{\frac{x \cdot z}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -1.18 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -1.18e-40)
     t_1
     (if (<= t 4e-34) (fma (- z t) (/ (- y x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - y), ((z - a) / t), y);
	double tmp;
	if (t <= -1.18e-40) {
		tmp = t_1;
	} else if (t <= 4e-34) {
		tmp = fma((z - t), ((y - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - y), Float64(Float64(z - a) / t), y)
	tmp = 0.0
	if (t <= -1.18e-40)
		tmp = t_1;
	elseif (t <= 4e-34)
		tmp = fma(Float64(z - t), Float64(Float64(y - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.18e-40], t$95$1, If[LessEqual[t, 4e-34], N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.1799999999999999e-40 or 3.99999999999999971e-34 < t

   1. Initial program 51.3%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   if -1.1799999999999999e-40 < t < 3.99999999999999971e-34

   1. Initial program 88.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{\left(y - x\right) \cdot \left(z - t\right)}{a} + x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z - t\right) \cdot \left(y - x\right)}}{a} + x \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{\left(z - t\right) \cdot \frac{y - x}{a}} + x \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z - t}, \frac{y - x}{a}, x\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{y - x}{a}}, x\right) \]

      7. lower--.f64 81.7

         \[\leadsto \mathsf{fma}\left(z - t, \frac{\color{blue}{y - x}}{a}, x\right) \]

   5. Applied rewrites 81.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -1.18 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -1.18e-40) t_1 (if (<= t 2.35e-21) (fma z (/ (- y x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - y), ((z - a) / t), y);
	double tmp;
	if (t <= -1.18e-40) {
		tmp = t_1;
	} else if (t <= 2.35e-21) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - y), Float64(Float64(z - a) / t), y)
	tmp = 0.0
	if (t <= -1.18e-40)
		tmp = t_1;
	elseif (t <= 2.35e-21)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.18e-40], t$95$1, If[LessEqual[t, 2.35e-21], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.1799999999999999e-40 or 2.35000000000000015e-21 < t

   1. Initial program 51.3%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Applied rewrites 75.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   if -1.1799999999999999e-40 < t < 2.35000000000000015e-21

   1. Initial program 87.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{y - x}{a}}, x\right) \]

      5. lower--.f64 78.8

         \[\leadsto \mathsf{fma}\left(z, \frac{\color{blue}{y - x}}{a}, x\right) \]

   5. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 66.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (- y x) (/ z (- a t)))))
   (if (<= z -1.7e+84)
     t_1
     (if (<= z 1.25e-26) (fma (- x y) (/ t (- a t)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - x) * (z / (a - t));
	double tmp;
	if (z <= -1.7e+84) {
		tmp = t_1;
	} else if (z <= 1.25e-26) {
		tmp = fma((x - y), (t / (a - t)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - x) * Float64(z / Float64(a - t)))
	tmp = 0.0
	if (z <= -1.7e+84)
		tmp = t_1;
	elseif (z <= 1.25e-26)
		tmp = fma(Float64(x - y), Float64(t / Float64(a - t)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.7e+84], t$95$1, If[LessEqual[z, 1.25e-26], N[(N[(x - y), $MachinePrecision] * N[(t / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6999999999999999e84 or 1.25000000000000005e-26 < z

   1. Initial program 68.3%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{a - t} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{z \cdot \color{blue}{\left(y - x\right)}}{a - t} \]

      6. lower--.f64 61.0

         \[\leadsto \frac{z \cdot \left(y - x\right)}{\color{blue}{a - t}} \]

   5. Applied rewrites 61.0%

      \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]
   6. Step-by-step derivation

   7. Applied rewrites 77.5%

      \[\leadsto \frac{z}{a - t} \cdot \color{blue}{\left(y - x\right)} \]

   if -1.6999999999999999e84 < z < 1.25000000000000005e-26

   1. Initial program 67.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + x} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{t \cdot \left(y - x\right)}{a - t}\right)\right)} + x \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot t}}{a - t}\right)\right) + x \]

      4. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{t}{a - t}}\right)\right) + x \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{t}{a - t}} + x \]

      6. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{t}{a - t} + x \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{t}{a - t}, x\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\left(y - x\right)\right)}, \frac{t}{a - t}, x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(y + \left(\mathsf{neg}\left(x\right)\right)\right)}\right), \frac{t}{a - t}, x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + y\right)}\right), \frac{t}{a - t}, x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)}, \frac{t}{a - t}, x\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - y}, \frac{t}{a - t}, x\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x} - y, \frac{t}{a - t}, x\right) \]

      14. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{x - y}, \frac{t}{a - t}, x\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - y, \color{blue}{\frac{t}{a - t}}, x\right) \]

      16. lower--.f64 68.3

          \[\leadsto \mathsf{fma}\left(x - y, \frac{t}{\color{blue}{a - t}}, x\right) \]

   5. Applied rewrites 68.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.7 \cdot 10^{+84}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-37}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{x \cdot z}{t}\\ \mathbf{elif}\;t \leq 28000000000:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -3.2e-37)
   y
   (if (<= t -3.5e-116)
     (/ (* x z) t)
     (if (<= t 28000000000.0) (* y (/ z a)) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.2e-37) {
		tmp = y;
	} else if (t <= -3.5e-116) {
		tmp = (x * z) / t;
	} else if (t <= 28000000000.0) {
		tmp = y * (z / a);
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-3.2d-37)) then
        tmp = y
    else if (t <= (-3.5d-116)) then
        tmp = (x * z) / t
    else if (t <= 28000000000.0d0) then
        tmp = y * (z / a)
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -3.2e-37) {
		tmp = y;
	} else if (t <= -3.5e-116) {
		tmp = (x * z) / t;
	} else if (t <= 28000000000.0) {
		tmp = y * (z / a);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -3.2e-37:
		tmp = y
	elif t <= -3.5e-116:
		tmp = (x * z) / t
	elif t <= 28000000000.0:
		tmp = y * (z / a)
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -3.2e-37)
		tmp = y;
	elseif (t <= -3.5e-116)
		tmp = Float64(Float64(x * z) / t);
	elseif (t <= 28000000000.0)
		tmp = Float64(y * Float64(z / a));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -3.2e-37)
		tmp = y;
	elseif (t <= -3.5e-116)
		tmp = (x * z) / t;
	elseif (t <= 28000000000.0)
		tmp = y * (z / a);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -3.2e-37], y, If[LessEqual[t, -3.5e-116], N[(N[(x * z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 28000000000.0], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], y]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.1999999999999999e-37 or 2.8e10 < t

   1. Initial program 50.6%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right) + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      3. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot z}}{a - t} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{\color{blue}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{a - t}, \color{blue}{-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + x}\right) \]

   5. Applied rewrites 80.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, \mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \left(x + -1 \cdot x\right) - \color{blue}{-1 \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.8%

      \[\leadsto y \]

   if -3.1999999999999999e-37 < t < -3.49999999999999984e-116

   1. Initial program 90.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{z}{t} - \frac{a}{t}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.0%

      \[\leadsto \frac{x \cdot \left(z - a\right)}{\color{blue}{t}} \]

   9. Taylor expanded in z around inf

      \[\leadsto \frac{x \cdot z}{t} \]
   10. Step-by-step derivation

   11. Applied rewrites 41.0%

       \[\leadsto \frac{z \cdot x}{t} \]

   if -3.49999999999999984e-116 < t < 2.8e10

   1. Initial program 86.4%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]
   4. Step-by-step derivation

      1. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot \left(y - x\right)}}{a - t} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{z \cdot \color{blue}{\left(y - x\right)}}{a - t} \]

      6. lower--.f64 50.1

         \[\leadsto \frac{z \cdot \left(y - x\right)}{\color{blue}{a - t}} \]

   5. Applied rewrites 50.1%

      \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} \]

   6. Taylor expanded in a around inf

      \[\leadsto \frac{z \cdot \left(y - x\right)}{\color{blue}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.0%

      \[\leadsto z \cdot \color{blue}{\frac{y - x}{a}} \]

   9. Taylor expanded in y around inf

      \[\leadsto \frac{y \cdot z}{a} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.0%

       \[\leadsto y \cdot \frac{z}{\color{blue}{a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-37}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{x \cdot z}{t}\\ \mathbf{elif}\;t \leq 28000000000:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z}{t}, y\right)\\ \mathbf{if}\;t \leq -1.18 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ z t) y)))
   (if (<= t -1.18e-40) t_1 (if (<= t 1.42e+82) (fma z (/ (- y x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - y), (z / t), y);
	double tmp;
	if (t <= -1.18e-40) {
		tmp = t_1;
	} else if (t <= 1.42e+82) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - y), Float64(z / t), y)
	tmp = 0.0
	if (t <= -1.18e-40)
		tmp = t_1;
	elseif (t <= 1.42e+82)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] * N[(z / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.18e-40], t$95$1, If[LessEqual[t, 1.42e+82], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.1799999999999999e-40 or 1.41999999999999993e82 < t

   1. Initial program 48.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(x - y, \frac{z}{\color{blue}{t}}, y\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 68.2%

      \[\leadsto \mathsf{fma}\left(x - y, \frac{z}{\color{blue}{t}}, y\right) \]

   if -1.1799999999999999e-40 < t < 1.41999999999999993e82

   1. Initial program 85.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{y - x}{a}}, x\right) \]

      5. lower--.f64 74.6

         \[\leadsto \mathsf{fma}\left(z, \frac{\color{blue}{y - x}}{a}, x\right) \]

   5. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 67.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \mathbf{if}\;t \leq -1.18 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.42 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ (- x y) t) y)))
   (if (<= t -1.18e-40) t_1 (if (<= t 1.42e+82) (fma z (/ (- y x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, ((x - y) / t), y);
	double tmp;
	if (t <= -1.18e-40) {
		tmp = t_1;
	} else if (t <= 1.42e+82) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(Float64(x - y) / t), y)
	tmp = 0.0
	if (t <= -1.18e-40)
		tmp = t_1;
	elseif (t <= 1.42e+82)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.18e-40], t$95$1, If[LessEqual[t, 1.42e+82], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.1799999999999999e-40 or 1.41999999999999993e82 < t

   1. Initial program 48.7%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Applied rewrites 78.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto y + \color{blue}{\frac{z \cdot \left(x - y\right)}{t}} \]
   7. Step-by-step derivation

   8. Applied rewrites 66.7%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x - y}{t}}, y\right) \]

   if -1.1799999999999999e-40 < t < 1.41999999999999993e82

   1. Initial program 85.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{x + \frac{z \cdot \left(y - x\right)}{a}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a} + x} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{a}} + x \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{y - x}{a}}, x\right) \]

      5. lower--.f64 74.6

         \[\leadsto \mathsf{fma}\left(z, \frac{\color{blue}{y - x}}{a}, x\right) \]

   5. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 57.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{x - y}{t}, y\right)\\ \mathbf{if}\;t \leq -2.05 \cdot 10^{-45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-40}:\\ \;\;\;\;x - \frac{x \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ (- x y) t) y)))
   (if (<= t -2.05e-45) t_1 (if (<= t 2.8e-40) (- x (/ (* x z) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, ((x - y) / t), y);
	double tmp;
	if (t <= -2.05e-45) {
		tmp = t_1;
	} else if (t <= 2.8e-40) {
		tmp = x - ((x * z) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(Float64(x - y) / t), y)
	tmp = 0.0
	if (t <= -2.05e-45)
		tmp = t_1;
	elseif (t <= 2.8e-40)
		tmp = Float64(x - Float64(Float64(x * z) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -2.05e-45], t$95$1, If[LessEqual[t, 2.8e-40], N[(x - N[(N[(x * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.05e-45 or 2.8e-40 < t

   1. Initial program 52.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Applied rewrites 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   6. Taylor expanded in a around 0

      \[\leadsto y + \color{blue}{\frac{z \cdot \left(x - y\right)}{t}} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.2%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{\frac{x - y}{t}}, y\right) \]

   if -2.05e-45 < t < 2.8e-40

   1. Initial program 88.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{z - t}{a - t}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{z - t}{a - t} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{z - t}{a - t}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{z - t}{a - t}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(z - t\right)}{a - t}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(z - t\right) \cdot x}}{a - t}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z - t\right) \cdot \frac{x}{a - t}}\right)\right) + 1 \cdot x \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right)} + 1 \cdot x \]

      10. *-lft-identity N/A

          \[\leadsto \left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right) + \color{blue}{x} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z - t}, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right) \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}\right)}, x\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}\right)}, x\right) \]

      17. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t} + \left(\mathsf{neg}\left(a\right)\right)}, x\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

      20. lower-neg.f64 63.0

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{t + \color{blue}{\left(-a\right)}}, x\right) \]

   5. Applied rewrites 63.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{x}{t + \left(-a\right)}, x\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto x + \color{blue}{-1 \cdot \frac{x \cdot z}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.8%

      \[\leadsto x - \color{blue}{\frac{x \cdot z}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 48.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \mathbf{if}\;t \leq -0.0075:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+81}:\\ \;\;\;\;x - \frac{x \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ (- y x) t) y)))
   (if (<= t -0.0075) t_1 (if (<= t 5.2e+81) (- x (/ (* x z) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, ((y - x) / t), y);
	double tmp;
	if (t <= -0.0075) {
		tmp = t_1;
	} else if (t <= 5.2e+81) {
		tmp = x - ((x * z) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(Float64(y - x) / t), y)
	tmp = 0.0
	if (t <= -0.0075)
		tmp = t_1;
	elseif (t <= 5.2e+81)
		tmp = Float64(x - Float64(Float64(x * z) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -0.0075], t$95$1, If[LessEqual[t, 5.2e+81], N[(x - N[(N[(x * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.0074999999999999997 or 5.19999999999999984e81 < t

   1. Initial program 45.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Applied rewrites 79.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto y + \color{blue}{-1 \cdot \frac{a \cdot \left(x - y\right)}{t}} \]
   7. Step-by-step derivation

   8. Applied rewrites 57.8%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{y - x}{t}}, y\right) \]

   if -0.0074999999999999997 < t < 5.19999999999999984e81

   1. Initial program 84.9%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{z - t}{a - t}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{z - t}{a - t} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{z - t}{a - t}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{z - t}{a - t}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(z - t\right)}{a - t}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(z - t\right) \cdot x}}{a - t}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z - t\right) \cdot \frac{x}{a - t}}\right)\right) + 1 \cdot x \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right)} + 1 \cdot x \]

      10. *-lft-identity N/A

          \[\leadsto \left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right) + \color{blue}{x} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z - t}, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right) \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}\right)}, x\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}\right)}, x\right) \]

      17. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t} + \left(\mathsf{neg}\left(a\right)\right)}, x\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

      20. lower-neg.f64 56.7

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{t + \color{blue}{\left(-a\right)}}, x\right) \]

   5. Applied rewrites 56.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{x}{t + \left(-a\right)}, x\right)} \]

   6. Taylor expanded in t around 0

      \[\leadsto x + \color{blue}{-1 \cdot \frac{x \cdot z}{a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.9%

      \[\leadsto x - \color{blue}{\frac{x \cdot z}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 33.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{+94}:\\ \;\;\;\;x \cdot \frac{z - a}{t}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+111}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{t - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -3.05e+94)
   (* x (/ (- z a) t))
   (if (<= x 1.05e+111) y (* x (/ z (- t a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -3.05e+94) {
		tmp = x * ((z - a) / t);
	} else if (x <= 1.05e+111) {
		tmp = y;
	} else {
		tmp = x * (z / (t - a));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= (-3.05d+94)) then
        tmp = x * ((z - a) / t)
    else if (x <= 1.05d+111) then
        tmp = y
    else
        tmp = x * (z / (t - a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -3.05e+94) {
		tmp = x * ((z - a) / t);
	} else if (x <= 1.05e+111) {
		tmp = y;
	} else {
		tmp = x * (z / (t - a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -3.05e+94:
		tmp = x * ((z - a) / t)
	elif x <= 1.05e+111:
		tmp = y
	else:
		tmp = x * (z / (t - a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -3.05e+94)
		tmp = Float64(x * Float64(Float64(z - a) / t));
	elseif (x <= 1.05e+111)
		tmp = y;
	else
		tmp = Float64(x * Float64(z / Float64(t - a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -3.05e+94)
		tmp = x * ((z - a) / t);
	elseif (x <= 1.05e+111)
		tmp = y;
	else
		tmp = x * (z / (t - a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -3.05e+94], N[(x * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.05e+111], y, N[(x * N[(z / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.05000000000000017e94

   1. Initial program 48.0%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{z}{t} - \frac{a}{t}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.0%

      \[\leadsto \frac{x \cdot \left(z - a\right)}{\color{blue}{t}} \]
   9. Step-by-step derivation

   10. Applied rewrites 53.0%

       \[\leadsto \frac{z - a}{t} \cdot x \]

   if -3.05000000000000017e94 < x < 1.04999999999999997e111

   1. Initial program 76.5%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right) + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      3. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot z}}{a - t} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{\color{blue}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{a - t}, \color{blue}{-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + x}\right) \]

   5. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, \mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \left(x + -1 \cdot x\right) - \color{blue}{-1 \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 34.8%

      \[\leadsto y \]

   if 1.04999999999999997e111 < x

   1. Initial program 59.8%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + -1 \cdot \frac{z - t}{a - t}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(-1 \cdot \frac{z - t}{a - t} + 1\right)} \]

      2. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{z - t}{a - t}\right) \cdot x + 1 \cdot x} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t}\right)\right)} \cdot x + 1 \cdot x \]

      4. distribute-lft-neg-out N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z - t}{a - t} \cdot x\right)\right)} + 1 \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{z - t}{a - t}}\right)\right) + 1 \cdot x \]

      6. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \left(z - t\right)}{a - t}}\right)\right) + 1 \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(z - t\right) \cdot x}}{a - t}\right)\right) + 1 \cdot x \]

      8. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(z - t\right) \cdot \frac{x}{a - t}}\right)\right) + 1 \cdot x \]

      9. distribute-rgt-neg-in N/A

         \[\leadsto \color{blue}{\left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right)} + 1 \cdot x \]

      10. *-lft-identity N/A

          \[\leadsto \left(z - t\right) \cdot \left(\mathsf{neg}\left(\frac{x}{a - t}\right)\right) + \color{blue}{x} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right)} \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z - t}, \mathsf{neg}\left(\frac{x}{a - t}\right), x\right) \]

      13. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \color{blue}{\frac{x}{\mathsf{neg}\left(\left(a - t\right)\right)}}, x\right) \]

      15. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}\right)}, x\right) \]

      16. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}\right)}, x\right) \]

      17. distribute-neg-in N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(t\right)\right)\right)\right) + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

      18. remove-double-neg N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t} + \left(\mathsf{neg}\left(a\right)\right)}, x\right) \]

      19. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{\color{blue}{t + \left(\mathsf{neg}\left(a\right)\right)}}, x\right) \]

      20. lower-neg.f64 71.9

          \[\leadsto \mathsf{fma}\left(z - t, \frac{x}{t + \color{blue}{\left(-a\right)}}, x\right) \]

   5. Applied rewrites 71.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z - t, \frac{x}{t + \left(-a\right)}, x\right)} \]

   6. Taylor expanded in z around inf

      \[\leadsto \frac{x \cdot z}{\color{blue}{t - a}} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.5%

      \[\leadsto x \cdot \color{blue}{\frac{z}{t - a}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.05 \cdot 10^{+94}:\\ \;\;\;\;x \cdot \frac{z - a}{t}\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{+111}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{t - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 28.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot z}{t}\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{+121}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* x z) t)))
   (if (<= x -8.2e+95) t_1 (if (<= x 1.52e+121) y t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * z) / t;
	double tmp;
	if (x <= -8.2e+95) {
		tmp = t_1;
	} else if (x <= 1.52e+121) {
		tmp = y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * z) / t
    if (x <= (-8.2d+95)) then
        tmp = t_1
    else if (x <= 1.52d+121) then
        tmp = y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * z) / t;
	double tmp;
	if (x <= -8.2e+95) {
		tmp = t_1;
	} else if (x <= 1.52e+121) {
		tmp = y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x * z) / t
	tmp = 0
	if x <= -8.2e+95:
		tmp = t_1
	elif x <= 1.52e+121:
		tmp = y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * z) / t)
	tmp = 0.0
	if (x <= -8.2e+95)
		tmp = t_1;
	elseif (x <= 1.52e+121)
		tmp = y;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * z) / t;
	tmp = 0.0;
	if (x <= -8.2e+95)
		tmp = t_1;
	elseif (x <= 1.52e+121)
		tmp = y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * z), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[x, -8.2e+95], t$95$1, If[LessEqual[x, 1.52e+121], y, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.19999999999999972e95 or 1.5199999999999999e121 < x

   1. Initial program 53.6%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   4. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto \color{blue}{y + \left(-1 \cdot \frac{z \cdot \left(y - x\right)}{t} - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      2. distribute-lft-out-- N/A

         \[\leadsto y + \color{blue}{-1 \cdot \left(\frac{z \cdot \left(y - x\right)}{t} - \frac{a \cdot \left(y - x\right)}{t}\right)} \]

      3. div-sub N/A

         \[\leadsto y + -1 \cdot \color{blue}{\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t} + y} \]

      5. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{z \cdot \left(y - x\right) - a \cdot \left(y - x\right)}{t}\right)\right)} + y \]

      6. distribute-rgt-out-- N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\left(y - x\right) \cdot \left(z - a\right)}}{t}\right)\right) + y \]

      7. associate-/l* N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(y - x\right) \cdot \frac{z - a}{t}}\right)\right) + y \]

      8. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\left(y - x\right)\right)\right) \cdot \frac{z - a}{t}} + y \]

      9. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \left(y - x\right)\right)} \cdot \frac{z - a}{t} + y \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(-1 \cdot \left(y - x\right), \frac{z - a}{t}, y\right)} \]

   5. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)} \]

   6. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(\frac{z}{t} - \frac{a}{t}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 37.7%

      \[\leadsto \frac{x \cdot \left(z - a\right)}{\color{blue}{t}} \]

   9. Taylor expanded in z around inf

      \[\leadsto \frac{x \cdot z}{t} \]
   10. Step-by-step derivation

   11. Applied rewrites 26.1%

       \[\leadsto \frac{z \cdot x}{t} \]

   if -8.19999999999999972e95 < x < 1.5199999999999999e121

   1. Initial program 76.2%

      \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)\right)} \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right) + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      3. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot z}}{a - t} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

      8. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{\color{blue}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{a - t}, \color{blue}{-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + x}\right) \]

   5. Applied rewrites 92.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, \mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \left(x + -1 \cdot x\right) - \color{blue}{-1 \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 34.4%

      \[\leadsto y \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+95}:\\ \;\;\;\;\frac{x \cdot z}{t}\\ \mathbf{elif}\;x \leq 1.52 \cdot 10^{+121}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 24.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ y \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 y)

C

double code(double x, double y, double z, double t, double a) {
	return y;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = y
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return y;
}

Python

def code(x, y, z, t, a):
	return y

Julia

function code(x, y, z, t, a)
	return y
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = y;
end

Wolfram

code[x_, y_, z_, t_, a_] := y

Derivation

1. Initial program 68.1%

   \[x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x + \left(-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)\right)} \]
4. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \color{blue}{\left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) + z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{z \cdot \left(\frac{y}{a - t} - \frac{x}{a - t}\right) + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

   3. div-sub N/A

      \[\leadsto z \cdot \color{blue}{\frac{y - x}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

   4. associate-/l* N/A

      \[\leadsto \color{blue}{\frac{z \cdot \left(y - x\right)}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(y - x\right) \cdot z}}{a - t} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

   6. associate-/l* N/A

      \[\leadsto \color{blue}{\left(y - x\right) \cdot \frac{z}{a - t}} + \left(x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right)} \]

   8. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{y - x}, \frac{z}{a - t}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

   9. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(y - x, \color{blue}{\frac{z}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

   10. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{\color{blue}{a - t}}, x + -1 \cdot \frac{t \cdot \left(y - x\right)}{a - t}\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(y - x, \frac{z}{a - t}, \color{blue}{-1 \cdot \frac{t \cdot \left(y - x\right)}{a - t} + x}\right) \]

5. Applied rewrites 88.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - x, \frac{z}{a - t}, \mathsf{fma}\left(x - y, \frac{t}{a - t}, x\right)\right)} \]

6. Taylor expanded in t around inf

   \[\leadsto \left(x + -1 \cdot x\right) - \color{blue}{-1 \cdot y} \]
7. Step-by-step derivation

8. Applied rewrites 25.0%

   \[\leadsto y \]
9. Add Preprocessing

Developer Target 1: 86.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))
   (if (< a -1.6153062845442575e-142)
     t_1
     (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (((y - x) / 1.0d0) * ((z - t) / (a - t)))
    if (a < (-1.6153062845442575d-142)) then
        tmp = t_1
    else if (a < 3.774403170083174d-182) then
        tmp = y - ((z / t) * (y - x))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)))
	tmp = 0
	if a < -1.6153062845442575e-142:
		tmp = t_1
	elif a < 3.774403170083174e-182:
		tmp = y - ((z / t) * (y - x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) / 1.0) * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = Float64(y - Float64(Float64(z / t) * Float64(y - x)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	tmp = 0.0;
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = y - ((z / t) * (y - x));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] / 1.0), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[a, -1.6153062845442575e-142], t$95$1, If[Less[a, 3.774403170083174e-182], N[(y - N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024233 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -646122513817703/4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 1887201585041587/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))))))

  (+ x (/ (* (- y x) (- z t)) (- a t))))